ANNIHILATION OF REAL CLASSES. Jean-Robert Belliard & Anthony Martin

Transcription

1 ANNIHILATION OF REAL CLASSES by Jean-Robert Belliard & Anthony Martin Abstract. Let F be a number field, abelian over Q and let p be a prime unramified in F. In this note we prove that Solomon s ψ F element annihilates the torsion of the Galois group of the maximal abelian p-ramified p-extension of F. This is another unconditional proven reinforcement of All theorem [All13] which was conjecture 4.1 of [Sol92]. 1. Introduction Let F be a number field, abelian over Q, and fix a prime p 2. For all n N fix ζ n a primitive n th root of unity in a coherent way, e.g. take ζ n = e 2iπ/n. Let f be the conductor of F, so that F Q(ζ f and f is minimal with respect to that property. Let X F denotes the p-part of the ideal class group of F. The Galois group F = = Gal(F/Q acts onto X F in a natural way, turning it into a Z p [ ]- module. One of the main challenge of algebraic number theory is to understand the structure of this object. As this is a finite module, a starting point would be to describe its annihilator or to find some canonical annihilators. If F is imaginary the Stickelberger element is such canonical element for the minus part of X F. However if F is real this Stickelberger element is a multiple of the absolute trace, hence the annihilation statement is trivial. Assume that F is totally real. Fix once and for all an embedding of Q in the field of complex number C and another embedding ı v : Q C p in the Tate field C p. This embedding uniquely defines a place v above p in F. Let F v C p be the closure of F and let O v be the local valuation ring at v. In [Sol92] Solomon defined an element ψ F O v [ ] as follows ψ F := 1 ( log p p ı v NQ(ζf /F (1 ζ f δ δ 1, δ F and stated the conjecture : Conjecture 1.1 ([Sol92], conjecture 4.1. ψ F annihilates X F O v. Let us state a few historical remarks, for which we need some more notations. Let X F be the Galois group of the maximal abelian p-ramified p-extension of F and let

2 2 JEAN-ROBERT BELLIARD & ANTHONY MARTIN tx F X F be its Z p -torsion module. Let also D F be the sub-module of X F generated the places dividing p. 1. If p (so called semi-simple case then conjecture 1.1 follows from the Main Conjecture of Iwasawa theory (here Mazur-Wiles theorem [MW84] : see remark 4.1 (ii of [Sol92]. 2. Theorem 4.1 of [Sol92] proves that ψ F annihilates D F O v. 3. If f is composite and if p is totally split in F, then theorem 5.4 of [BNQD05] proves conjecture In full generality the theorem 1.1 of [All13] is a proven reinforcement of conjecture 1.1. The point here is to construct global explicit annihilators of all X F O v inside Z p [ ] without using χ-eigenspaces, so the semi-simple cases are not relevant to this problem. The main result of this note is Theorem 1.2. ψ F annihilates tx F O v. As tx F maps surjectively onto X F, this theorem is another reinforcement of conjecture 1.1. The theorem 1.2 reinforces conjecture 1.1, but it also proves that the element ψ F is not a real analogue of Stickleberger element, in the sense that ψ F is more naturally associated to tx F than to X F. The interested reader should consult [NQDN11] and the pioneering work [Sol09] where a better real analogue of Stickelberger ideal are introduced and studied. The strategy to prove theorem 1.2 is quite simple. One first use Iwasawa theory to consider objects X and X defined by replacing F with its cyclotomic Z p -extension F. Then one use Coleman theory to define a global power series Col(η f O v [ ][[T ]] such that ψ F is recovered from Col(η f simply by setting T = 0. Now by classical Iwasawa s Main Conjecture, which is a statement about χ-parts, all χ components of Col(η f annihilates all χ-parts of X. But the point here is that X has simultaneously no finite sub-module and trivial µ-invariant, contrary to X which is conjectured to be finite. Therefore the global elements Col(η f actually annihilates full X and theorem 1.2 follows by taking co-invariants. 2. Setting Let us recall the analytic and algebraic setting of Iwasawa Main Conjecture Galois setting. Fix once and for all a number field F, abelian over Q, real, and of conductor f. Fix a prime number p 2 such that p f. For all integer n, the field F n will be the n-th step of the cyclotomic Z p -extension of F, so that [F n : F ] = p n. Also we put K n = F (ζ p n and Kn = Gal(K n /Q, Fn = Gal(F n /Q. As p f, we have K n = F n (ζ p. Set also F = n F n and K = n K n. By disjoint ramification the relevant extensions are split and we have direct product of various

3 ANNIHILATION OF REAL CLASSES 3 Galois groups as follows : Fn F Gal(F n /F F Z/p n Z; Kn F Gal(K/F Gal(K n /K F Z/(p 1Z Z/p n Z. The whole Galois setting is summarized as follows K F K n F n F F (µ p = K Q(fp Q(fp n Q(fp Q. For any finite abelian group G and any valuation ring O of a finite extension of Q p, we will denote by Λ O [G] the Iwasawa algebra Λ O [G] = (lim O[Z/p n Z][G] O[G][[T ]]. If O = Z p we simply note Λ[G] = Λ O [G]. Moreover we will note ω the Teichmüller character defined by the embedding ı v. Precisely we have for all a in Z either p a and ω(a = 0 either (a, p = 1 and ω(a is the unique (p 1 th root of unity (simultaneously in C and C p such that v(ω(a a > 0. Identifying Gal(Q(ζ p /Q with (Z/pZ, this character describes the action of Galois on the p th -power root of unity as for all g and all such root of unity ζ we have g(ζ = ζ ω(g, the right hand side being well defined because ω(g Z p. We also will denote by κ: Gal(Q(fp /Q(f Z p the canonical isomorphism (abusively called cyclotomic character. For all a in Z p we will denote σ a the element of Gal ( Q(ζ fp Q such that κ(σ a = a Analytic side. The p-adic L function is defined by interpolating the values of the complex L functions at negative integers. To be more precise let us recall a fex classical notations. Let χ be a Dirichlet character, assumed to be primitive modulo f. Via the Artin map the character χ may be seen as a Galois Character, actually a character of Gal(Q(ζ f Q. Let Q(χ Q be the field generated over Q by the values of χ. The χ-twisted Bernoulli numbers B n,χ Q(χ are defined by

4 4 JEAN-ROBERT BELLIARD & ANTHONY MARTIN the power series expansion f χ(ase as e fs 1 = s n B n,χ n!. a=1 The complex L function L(s, χ takes algebraic values at negative integers. Indeed according to [Was97] theorem (1 one has : L(1 n, χ = B n,χ n. Now by making use of the fixed embedding ı v it is possible to interpolate these values with a C p valued function. Theorem 2.1 (cf. [Was97] theorem Exists a unique meromorphic function L p (., χ (which is holomorphic if f 1 defined on the open ball {s C p s < pp 1/p 1 } and such that for all n 1 : L p (1 n, χ = (1 χω n (pp n 1 B n,χω n n n=0 = (1 χω n (pp n 1 L(1 n, χω n. Remark 2.2. If f 1, then one has the Leopoldt formula (see [Was97] theorem 5.18 ( L p (1, χ = 1 χ(p τ(χ f χ(a 1 log p f p ı v (1 ζf a, where τ = τ(χ = a=f a=1 χ(aζa f is the Gauß sum associated to χ. Therefore the Solomon s element ψ F is obviously related to these values at 1. These p-adic L functions are uniquely determined by this interpolation properties. Their original definition using Mahler development of partial zeta functions does not fully reveal their real nature. Later on Iwasawa discovered a far more concrete definition by using a sequence of Stickelberger elements to obtain a power series development of these functions. These power series are elements of Z p (χ[[t ]]. Definition 2.3. The Stickelberger element ξ related to Q(ζ Qζfp n fpn is defined by the formula ξ Q(ζfp n = 1 a=fp n 1 ( fp n fp n 2 a σa 1 Q[Gal(Q(ζ fp n/q]. (a,fp=1;a=1 By definition the Stickelberger element related to K n is the element deduced from ξ Q(ζfp n by restriction from Gal(Q(ζ fp n/q to Kn. ξ Kn = 1 fp n One can then show that (1 completer la ref a=fp n 1 (a,fp=1;a=1 a=1 ( fp n 2 a (σa 1 Kn Q[ Kn ].

5 ANNIHILATION OF REAL CLASSES 5 1. Let ϑ Q(ζfp n = ( 1 (1 + fp(κ 1 (1 + fp 1 ξ Q(ζfp n. Then one has ϑ Q(ζfp n Z p [Gal(Q(ζ fp n/q], so that its restriction ϑ Kn to Q[ Kn ] is also a p-adic integral element actually an element of Z p [ Kn ] 2. The sequences (ξ Kn n N and therefore (ϑ Kn n N are coherent with respect to restrictions maps along the towers Q(fp /Q(f and K /K. As the ϑ Kn are coherent along K /K we may consider the limit ϑ K Λ( K. Let χ = χ 1 ω be the mirror character of χ. Then χ is a Dirichlet character of K. As all other character of K our χ extends linearly to a ring homomorphism χ : Λ( K Λ(χ, where Λ(χ is just the ring obtained by adjoining to Λ the values of χ. Let us fix the topological generator γ = κ 1 (1 + fp of Γ = Gal(K /K. Then sending γ to 1 + T uniquely define the Serre s isomorphism S : Λ(χ Z p (χ[[t ]]. Definition 2.4. We may now define the three relevant Iwasawa s power series attached to our Dirichlet character χ : 1. g(t, χ = S(χ (ϑ K Z p (χ[[t ]]; 2. h(t, χ = pf 1 + T Z p[[t ]]; g(t, χ 3. f(t, χ = h(t, χ Z p(χ[[t ]]. Iwasawa in [Iwa69] discovered that these series would provide another definition of the functions L p (s, χ. This definition turned out to be far more enlightening than the original one. Theorem 2.5 ([Iwa69]. Let χ be an even character of K. Then for all s C p such that s < pp 1/(p 1 (and s 1 if χ is trivial we have L p (s, χ = f((1 + fp s 1, χ Algebraic side. For all number field F recall that X F is the p-part of the class group of F. X k is the Galois group of the maximal abelian p-ramified p-extension of F. We X = lim X Fn, X = lim X Fn (with respect to norms and restriction maps. These are finitely generated Λ( F -modules and are linked together by the exact sequence where 0 E U X X 0 E is the direct limit of the E n = E n Z p where the E n are the global units of the F n. U is the direct limit of the semi-local units of the F n.

6 6 JEAN-ROBERT BELLIARD & ANTHONY MARTIN By the weak Leopoldt conjecture (here a theorem the Λ-module X is torsion. In particular, for all character χ of F, the χ-part X χ is a finitely generated torsion Z p (χ[[t ]]-module. We therefore may consider its characteristic series that we will denote c.s.(x χ. Remark 2.6. It is a well known theorem of Iwasawa (see [Iwa59] that X χ has no non-trivial finite Λ-submodule. In particular it is annihilated by its characteristic series Main conjecture. The main conjecture of Iwasawa relevant to this context may be stated as follows : Theorem 2.7 (Mazur-Wiles. For all character χ of F we have s.c.(x χ (T = f (T, χ where, by definition, the Iwasawa s involution is given by ( 1 + fp f (T, χ = f 1 + T 1, χ. The mirror Stickelberger series f (T, χ plays an important role here. Another (equivalent statement of the main conjecture is the following equality of characteristic series : s.c.(x ψ (T = s.c.(u /C ψ (T, where C stands by definition for the direct limit of circular units of the F n. 3. Solomon s element Recall that for an abelian totally real field F such that p is unramified in F the Solomon s element is defined by the formula ψ F := 1 ( log p p ıv (N Q(ζf /F (1 ζ f δ δ 1. δ F Our goal in that section is to use Coleman theory to relate this ψ F to the constant term of a power series that annihilates X Coleman s theory. This theory is essentially a local one and it stands for any unramified extension of Q p. Let O v [[X]] be the ring of formal power series in one indeterminate X with coefficient in O v. It is important here not to mix the Coleman X with the Iwasawa s T. Let O v ((X be the field of fraction of O v [[X]]. We will also denote Γ = Gal(Q p (ζ p /Q p ; this notation is coherent with the previous notation Γ = Gal(K /K. The rings O v [[X]] and O v ((X are endowed with : 1. the action of a Frobenius ϕ such that f ϕ (X = f((1 + X p a norm operator N uniquely determined by N (f ϕ (X = ξ µ p f(ξ(1+x 1.

7 ANNIHILATION OF REAL CLASSES 7 3. an action of Z p [[Γ ]] which extends by linearity and continuity the action of Γ defined by γ.(1 + X = (1 + X κ(γ for all γ Γ (recall that κ is the cyclotomic character. Theorem 3.1 (Coleman. Let (α n = α lim F v (ζ p n be a norm coherent sequence of local numbers. Then there exists a unique power series f α O v ((X such that f α (ζ p n+1 1 = α n and N f α = f ϕ α. Let Col : lim F v (ζ p n O v ((X be the map defined by Col(α = (1 ϕ p log p f α (X = 1 p log p Then Col induces an exact sequence where 1 Z p (1 U 1 ( fα (X p f ϕ α (X. Col R ɛ Z p (1 0. U 1 is the direct limit of the principal units of F v (ζ p n; R is a rank one free O v [[Γ ]]-module generated by (1 + X; Z p (1 is the direct limit of the (ζ p n; ɛ is the map f Df X=0 where D = (1 + X d dx. It is useful to consider another related exact sequence that is obtained by using the Mellin transform L of Col with values in O v [[Γ ]] instead of R. Let us briefly restate this definition : We saw that R is a rank one free O v [[Γ ]] generated by (1 + X. Hence all element g R may uniquely be written as g(x = ĝ.(1 + X with ĝ Z p [[Γ ]]. The element ĝ (also denoted by Mel(g is called the Mellin transform of g and we define L = Mel Col Formulas. We need to give a dictionary of formulas between R and O v [[Γ ]]. For all ĝ = a n γ n O v [[Γ ]] and all characters ψ of Γ we note ψ(ĝ = a n ψ(γ n Z p (ψ. Recall that an γ n. (1 + X = a n (1 + X κn (γ where κ is the cyclotomic character. It follows that for all g R, i Z g(0 = 1 ( ĝ (where 1 stands for the trivial character. D i g = T w i( ĝ, where if ĝ = a n γ n then T w i( ĝ = a n (κ i (γγ n One has in particular ( D i g X=0 = 1( T w i (ĝ = κ i (ĝ. With all these formulas and notation the twin sequence of the exact sequence of Coleman is now (1 1 Z p (1 U 1 L O v [[Γ κ ]] Z p (1 0.

8 8 JEAN-ROBERT BELLIARD & ANTHONY MARTIN Induction. Let L be the splitting field at p of F, so that p is totally split in L/Q and that primes above p remains inert in F/L. In order to recover global information for our field F from the local information collected in F v one first need to induce along Gal(L/Q and consider first the semi-local situation. We have Gal(F v /Q p Gal(F/L. Let us abbreviate L = Gal(L/Q. We apply the (exact functor Zp[Gal(F/L] Z p [ F ] to the sequence (1. We get (see also[tsu99] : (2 0 Z p [ L ](1 U ÔF [[Γ ]] Z p [ L ](1 0, where ÔF := O F Z p Z p [ F ] O v Zp[Gal(F/L] Z p [ F ] O v [ L ] because O v is Galois free (recall that p is unramified in F cyclotomic units and Solomon s element. Consider the cyclotomic elements η Fn := N Q(ζfp n/f n (1 ζ fp n. These numbers are cyclotomic units of F n (recall that f 1 and they form a norm coherent sequence (this follows from the well known distribution relation satisfied by the numbers 1 ζ n. This allows us to define : Definition 3.2. Soit η F, = ( N Q(ζfp n/f n (1 ζ fp n. n N This is an element of lim O F n that may be seen as an element of U using the diagonal embedding. We can now state and prove the key result of this note : Theorem 3.3. The Solomon s element ψ F is nothing more than ψ F = 1 p 1 1 (L(η F,. Proof. By the formulas given in we have 1 ( L(η F = Col ( η F, (0, so that the theorem is equivalent to the identity (p 1ψ F = Col ( η F, (0. We shall check this equality. First remark that the set of p-places of F is in one to one correspondence with L via the map L {v, v is above p} δ δ 1 v Hence for all δ L, the δ-component of η F, (seen as an element of U is ı v (N Q(ζfp n/f n (1 ζ δ 1 f ζ p n n. Therefore we locally have (at the corresponding place δ 1 v that the associated Coleman series is ( ι v 1 ζf σδ 1 (1 + X. σ Gal(Q(ζ f /F

9 ANNIHILATION OF REAL CLASSES 9 Therefore, and still at the place δ 1 v, we obtain ( Col(η f, = 1 ϕ f ηf, (X p = 1 ( 1 ζ σδ 1 p log f (1 + X p p ( 1 ζ σδ 1 f (1 + X p It follows that specializing at X = 0 we find that the δ-component of Col(η f, (0 is p 1 log p p ι v N Q(ζf /F (1 ζf δ 1 which is exactly (p 1 times the δ-component of ψ F. We now prove that the power series related to η f, annihilates X. Lemma 3.4. The element L(η F, annihilates X O v. Proof. By the main conjecture, for all character χ of F, the characteristic series of X χ (which annihilates X χ is the same as the one of ( χ. U /C But for all ( character χ of F the series χ(l(η F is a multiple of the characteristic series of χ. U /C This proves that χ(l(ηf annihilates X χ for all χ. But the whole X has no non-trivial p-torsion (Ferrero-Washington and Iwasawa for the finite Λ- module, [FW79, Iwa59]. The result follows. Now our main result is Theorem 3.5. The Solomon elements ψ F annihilates the Z p -torsion tx F of X F. Proof. Putting together the theorem 3.3 and the lemma 3.4 and Iwasawa coinvariant we see that (p 1ψ F annihilates (X Γ. But these Iwsawa co-invariant are known to be canonically isomorphic to tx F. Remark 3.6. It is an interesting but technically difficult question to precisely relate the global integral Coleman power series L(η F, with the global integral power series ϑ K Λ( K and also to relate both to the Deligne-Ribet pseudomeasure. All these objects seems to be various avatar of the p-adic L function but there is some discrepancy between them. We hope to be able to say more in a future work. References [All13] Timothy All, On p-adic annihilators of real ideal classes, J. Number Theory 133 (2013, no. 7, MR [BNQD05] Jean-Robert Belliard and Thong Nguyen-Quang-Do, On modified circular units and annihilation of real classes, Nagoya Math. J. 177 (2005, [FW79] Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant µ p vanishes for abelian number fields, Ann. of Math. (2 109 (1979, no. 2, MR (81a:12005 [Iwa59] Kenkichi Iwasawa, On the theory of cyclotomic fields, Ann. of Math. (2 70 (1959, MR (23 #A1634

10 10 JEAN-ROBERT BELLIARD & ANTHONY MARTIN [Iwa69], On p-adic L-functions, Ann. of Math. (2 89 (1969, MR (42 #4522 [MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984, no. 2, [NQDN11] Thong Nguyen Quang Do and Vésale Nicolas, Nombres de Weil, sommes de Gauss et annulateurs galoisiens, Amer. J. Math. 133 (2011, no. 6, MR [Sol92] David Solomon, On a construction of p-units in abelian fields, Invent. Math. 109 (1992, no. 2, [Sol09], Some New Maps and Ideals in Classical Iwasawa Theory with Applications, arxiv: (2009, 36. [Tsu99] Takae Tsuji, Semi-local units modulo cyclotomic units, J. Number Theory 78 (1999, no. 1, MR (2000f:11148 [Was97] Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, MR (97h:11130 March 23, 2014 Jean-Robert Belliard, Laboratoire de mathématiques de Besançon UMR 6623, Université de Franche-Comté, 16 route de Gray, Besançon cedex, FRANCE jean-robert.belliard@univ-fcomte.fr Anthony Martin, Laboratoire de mathématiques de Besançon UMR 6623, Université de Franche-Comté, 16 route de Gray, Besançon cedex, FRANCE anthony.martin@ens-lyon.org